Write a polynomial for the area between two concentric circles if the radius of the outer circle is two less than three times the radium of the inner circle.
To find the polynomial for the area between two concentric circles, we first need to express the radii of the inner and outer circles in terms of a variable, such as "x."
Let's assume the radius of the inner circle is "r." According to the problem statement, the radius of the outer circle is two less than three times the radius of the inner circle. Therefore, the radius of the outer circle would be: 3r - 2.
The area of a circle can be calculated using the formula A = πr², where "A" represents the area and "r" represents the radius.
Now, let's find the area between the two circles by subtracting the area of the inner circle from the area of the outer circle:
A_outer = π(3r - 2)²
A_inner = πr²
Area_between_circles = A_outer - A_inner
= π(3r - 2)² - πr²
Simplifying this expression further, we expand the square:
Area_between_circles = π(9r² - 12r + 4) - πr²
Now, we can simplify this polynomial further:
Area_between_circles = 9πr² - 12πr + 4π - πr²
= (9π - π)r² - 12πr + 4π
= 8πr² - 12πr + 4π
Therefore, the polynomial for the area between the two concentric circles is: 8πr² - 12πr + 4π.